oppc1stf - Mathbox for Zhi Wang

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Theorem oppc1stf 50517

Description: The opposite functor of the first projection functor is the first projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)

**Hypotheses**
Ref	Expression
oppc1stf.o	⊢ 𝑂 = (oppCat‘𝐶)
oppc1stf.p	⊢ 𝑃 = (oppCat‘𝐷)
oppc1stf.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
oppc1stf.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

**Assertion**
Ref	Expression
oppc1stf	⊢ (𝜑 → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = (𝑂 1^st_F 𝑃))

Proof of Theorem oppc1stf Dummy variables 𝑥 𝑦 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppc1stf.o	. 2 ⊢ 𝑂 = (oppCat‘𝐶)
2		oppc1stf.p	. 2 ⊢ 𝑃 = (oppCat‘𝐷)
3		oppc1stf.c	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
4		oppc1stf.d	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
5		eqid 2760	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))
6	5	tposmpo 8285	. . . . 5 ⊢ tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))
7		eqid 2760	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	7, 1	oppchom 17896	. . . . . . . . 9 ⊢ ((1^st ‘𝑦)(Hom ‘𝑂)(1^st ‘𝑥)) = ((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦))
9		eqid 2760	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	9, 2	oppchom 17896	. . . . . . . . 9 ⊢ ((2^nd ‘𝑦)(Hom ‘𝑃)(2^nd ‘𝑥)) = ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦))
11	8, 10	xpeq12i 5678	. . . . . . . 8 ⊢ (((1^st ‘𝑦)(Hom ‘𝑂)(1^st ‘𝑥)) × ((2^nd ‘𝑦)(Hom ‘𝑃)(2^nd ‘𝑥))) = (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))
12		eqid 2760	. . . . . . . . 9 ⊢ (𝑂 ×_c 𝑃) = (𝑂 ×_c 𝑃)
13		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	1, 13	oppcbas 17899	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝑂)
15		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	2, 15	oppcbas 17899	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝑃)
17	12, 14, 16	xpcbas 18359	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝑂 ×_c 𝑃))
18		eqid 2760	. . . . . . . . 9 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
19		eqid 2760	. . . . . . . . 9 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
20		eqid 2760	. . . . . . . . 9 ⊢ (Hom ‘(𝑂 ×_c 𝑃)) = (Hom ‘(𝑂 ×_c 𝑃))
21		simp2 1155	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))
22		simp3 1156	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
23	12, 17, 18, 19, 20, 21, 22	xpchom 18361	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥) = (((1^st ‘𝑦)(Hom ‘𝑂)(1^st ‘𝑥)) × ((2^nd ‘𝑦)(Hom ‘𝑃)(2^nd ‘𝑥))))
24		eqid 2760	. . . . . . . . 9 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
25	24, 13, 15	xpcbas 18359	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×_c 𝐷))
26		eqid 2760	. . . . . . . . 9 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
27	24, 25, 7, 9, 26, 22, 21	xpchom 18361	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦) = (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦))))
28	11, 23, 27	3eqtr4a 2821	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥) = (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))
29	28	reseq2d 5970	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥)) = (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))
30	29	mpoeq3dva 7499	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))))
31	6, 30	eqtr4id 2814	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥))))
32	31	opeq2d 4846	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))⟩ = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥)))⟩)
33		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝐶 ∈ Cat)
34		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝐷 ∈ Cat)
35		eqid 2760	. . . . 5 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
36	24, 25, 26, 33, 34, 35	1stfval 18372	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝐶 1^st_F 𝐷) = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))⟩)
37	24, 33, 34, 35	1stfcl 18378	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝐶 1^st_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐶))
38	36, 37	oppfval3 50361	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))⟩)
39	1	oppccat 17903	. . . . 5 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
40	33, 39	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑂 ∈ Cat)
41	2	oppccat 17903	. . . . 5 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
42	34, 41	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑃 ∈ Cat)
43		eqid 2760	. . . 4 ⊢ (𝑂 1^st_F 𝑃) = (𝑂 1^st_F 𝑃)
44	12, 17, 20, 40, 42, 43	1stfval 18372	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝑂 1^st_F 𝑃) = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥)))⟩)
45	32, 38, 44	3eqtr4d 2805	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = (𝑂 1^st_F 𝑃))
46		df-1stf 18354	. 2 ⊢ 1^st_F = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⦋((Base‘𝑐) × (Base‘𝑑)) / 𝑏⦌⟨(1^st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1^st ↾ (𝑥(Hom ‘(𝑐 ×_c 𝑑))𝑦)))⟩)
47	1, 2, 3, 4, 45, 46	oppc1stflem 50516	1 ⊢ (𝜑 → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = (𝑂 1^st_F 𝑃))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ⦋csb 3847 ⟨cop 4834 × cxp 5648 ↾ cres 5652 ‘cfv 6541 (class class class)co 7422 ∈ cmpo 7424 1^st c1st 8001 2^nd c2nd 8002 tpos ctpos 8247 Basecbs 17394 Hom chom 17446 Catccat 17845 oppCatcoppc 17892 ×_c cxpc 18349 1^st_F c1stf 18350 oppFunc coppf 50345

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-om 7880 df-1st 8003 df-2nd 8004 df-tpos 8248 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-er 8722 df-map 8854 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-z 12699 df-dec 12820 df-uz 12971 df-fz 13645 df-struct 17332 df-sets 17349 df-slot 17367 df-ndx 17379 df-base 17395 df-hom 17459 df-cco 17460 df-cat 17849 df-cid 17850 df-homf 17851 df-comf 17852 df-oppc 17893 df-func 18040 df-xpc 18353 df-1stf 18354 df-oppf 50346

This theorem is used by: (None)

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